Linear Momentum: Mass in Motion

Which is harder to stop?

Picture two things sliding toward you on the ice at the local arena. One is a curling stone gliding at a slow walking pace. The other is a hockey puck flying off a slapshot. The puck is much faster. But most people would much rather stop the puck than the stone. Why?

Stopping something depends on both its mass and its velocity. Physicists combine them into one quantity called linear momentum. Newton called it the quantity of motion. It is the star of this whole unit: in every collision, explosion and rocket launch you will study, momentum is the quantity that is shared, traded and conserved.

Remember from Grade 11: in SPH3U you used Newton's laws (F_net = ma, action-reaction pairs) and kinetic energy, Ek = ½mv². In the sibling unit Work, Energy and Simple Harmonic Motion you extended work and energy to two dimensions and added elastic potential energy. Momentum is new: this unit builds it from the ground up and then uses it alongside energy.

Defining momentum

The linear momentum of an object is its mass times its velocity:

p = mv

• p is momentum, in kilogram metres per second (kg·m/s).
• m is mass in kilograms, and v is velocity in metres per second.

Because velocity is a vector, momentum is a vector too. It points in the same direction as the velocity. A 2.0 kg cart moving 3.0 m/s east has a momentum of 6.0 kg·m/s east. The same cart moving 3.0 m/s west has 6.0 kg·m/s west. Those two momenta are not equal, even though their sizes match.

In one dimension, we handle direction with signs. Choose one direction as positive (usually east, right or up). Then a velocity or momentum the other way is negative. Always state your sign convention at the start of a problem.

The word linear separates this from angular momentum (the momentum of spinning objects), which you will meet in university physics. In this unit, "momentum" always means linear momentum.

Momentum and kinetic energy: two different ideas

Momentum and kinetic energy both depend on mass and speed, so students often mix them up. They are very different:

• Momentum p = mv is a vector. It depends on speed to the first power. Two objects moving in opposite directions can have momenta that cancel.
• Kinetic energy Ek = ½mv² is a scalar. It depends on speed squared. It is never negative, so kinetic energies never cancel.

You can link the two. Since v = p/m,

Ek = ½m(p/m)² = p² ÷ (2m), or equivalently p = √(2mEk).

This tells you something useful: if two objects have the same momentum, the lighter one has more kinetic energy. You will use this fact in explosions, where the lighter fragment always gets more of the energy.

Momentum of a system, and change in momentum

A system is the group of objects you choose to study, such as two carts on a track or two players on the ice. The total momentum of a system is the vector sum of the momenta of all its parts:

p_total = m₁v₁ + m₂v₂ + ...

In one dimension, add with signs. In two dimensions (later in this unit), add vectors by components.

The change in momentum of an object is Δp = p_f − p_i (final minus initial). Because this is a vector subtraction, a ball that bounces back has a much bigger change in momentum than one that just stops: reversing direction means losing all your momentum and gaining momentum the other way.